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Open research questions in Analytic and geometric function theory

59 unresolved questions extracted from the limitations and future-work sections of 466 Analytic and geometric function theory papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • To extend the results of the paper to higher dimensions. To compare the performance of the proposed method with other methods. To apply the results of the paper to other areas of mathematics and computer science.

    On the Banach-Mazur Ellipse · 2026 · DOI

  • The authors suggest that future research should focus on extending the results to a wider range of h. The paper proposes that future work should investigate the application of the technique for averaging the tropical distance function over the space of all tropical structures of fixed co-area to other fields. The authors recommend that future research should explore the implications of the paper's results for the study of convex domains and their equi-affine invariants.

    Limits of Equi-Affine Equi-Distant Loci of Planar Convex Domains with Two Non-Parallel Asymptotes · 2026 · DOI

  • The paper identifies a gap in the field of equi-affine geometry regarding the study of convex domains and their equi-affine invariants. The authors note that the prior work on tropical structures and their application to convex domains is limited. The paper aims to fill this gap by introducing a family of equi-affine invariant functions associated with convex domains.

    Limits of Equi-Affine Equi-Distant Loci of Planar Convex Domains with Two Non-Parallel Asymptotes · 2026 · DOI

  • The study of non-closed mappings is identified as a gap in the existing literature. The connection between Orlicz-Sobolev classes and estimates of the distortion of the modulus of families of paths has not been fully established.

    ON BOUNDARY EXTENSION OF UNCLOSED ORLICZ–SOBOLEV MAPPINGS · 2026 · DOI

  • The need for an equivalent characterization for a subclass of Robertson functions - The lack of sharp upper bounds of the Schwarzian norm for the subclass S α

    Pre-Schwarzian and Schwarzian norm estimates for Robertson class · 2026 · DOI

  • The problem of finding the minimal Banach-Mazur distance to the Euclidean space is a fundamental problem in geometry of normed spaces, but it is not well understood. The existing methods for estimating the Banach-Mazur distance may not be optimal in high dimensions.

    On the Banach-Mazur Ellipse · 2026 · DOI

  • Prior methods have limitations, such as unwieldy calculations or complex residues.

    Determination of a Driving-Point Function from its Specified Real Part · 1983 · DOI

  • The explicit constants Ch and Cs in Lemma 4.3 are computed for the specific geometry of a disk domain with radius δ. The paper does not provide a systematic method for computing corresponding constants for other bounded domains (e.g., ellipses, polygonal regions), which would be necessary for applying the bi-Lipschitz framework to practical geometric settings.

    The bi-Lipschitz constant of an isothermal coordinate chart · 2026 · DOI

  • The proof of Theorem 1.1 uses Liouville's equation (3) under the assumption of a smooth conformal factor φ, but the paper does not discuss how the bi-Lipschitz bounds behave when φ has discontinuities or when the isothermal coordinate chart degenerates. Validation of the bounds under weakened regularity conditions would extend the theoretical framework.

    The bi-Lipschitz constant of an isothermal coordinate chart · 2026 · DOI

  • If α1 + α2 → 0, then A1 → 0, and the parameters ρ and τ are no longer well defined. Therefore, such limiting cases fall outside the scope of Theorem 3.4 and are excluded from consideration.

    On a certain subclass of strongly starlike functions · 2026 · DOI

  • The lack of estimates for coefficient bounds of regular functions using a q-series approach. The need for a novel set of regular q-functions to study complex functions.

    Four-Leaf Inequalities: A <i>q</i>-Series Approach · 2026 · DOI

  • The traditional approach to generating rational approximants of irrational numbers has limitations. There is a need for a new approach to analyzing continued fractions in hyperbolic space.

    Continued fractions and distorted chequer boards · 2026 · DOI

  • To study the behavior of anisotropic materials using the results of the paper. To optimize the shape of objects with respect to a surface tension using the results of the paper. To study the behavior of biological systems with anisotropic properties using the results of the paper.

    On the Anisotropic Partitioning Problem in Euclidean Convex Domains · 2026 · DOI

  • The lack of a complete description of the solutions to the anisotropic partitioning problem. The need for sharp isoperimetric inequalities with respect to convex cones. The need for a second variation formula for the anisotropic area of hypersurfaces with non-empty boundary.

    On the Anisotropic Partitioning Problem in Euclidean Convex Domains · 2026 · DOI

  • The lack of sharp coefficient bounds and Hankel determinants for the class S * cos. The need for a deeper understanding of starlike functions and their applications.

    Sharp inverse logarithmic coefficient bounds for starlike functions associated with cosine function · 2026 · DOI

  • The paper does not provide a comprehensive comparison with other existing methods. The results are limited to the subclass of analytic functions defined on a petal-shaped domain.

    Analysis of certain determinants for a defined subclass of analytic functions using Poisson distribution series in petal shaped domain · 2026 · DOI

  • The paper suggests further research into coefficient functional problems related to Poisson distribution series of analytic functions. The paper proposes the study of other subclasses of analytic functions defined on different domains.

    Analysis of certain determinants for a defined subclass of analytic functions using Poisson distribution series in petal shaped domain · 2026 · DOI

  • The study of coefficient problems within the novel subclass is identified as a research gap. The analysis of Hankel determinants for the subclass is also identified as a gap.

    Properties of some univalent functions associated with balloon-shaped domain · 2026 · DOI

  • The paper identifies a gap in the understanding of the oscillation of Fourier coefficients of automorphic forms. The paper aims to fill this gap by establishing a quantitative result regarding the sign changes of the sequence of Fourier coefficients.

    ON THE QUANTITATIVE SIMULTANEOUS SIGN CHANGES OF FOURIER COEFFICIENTS ATTACHED TO HECKE-MAASS FORMS OVER CERTAIN QUADRATIC FORMS · 2026 · DOI

  • To investigate whether the results can be extended to other generalized growth scales and growth-oscillation frameworks. To explore the relationships between different growth theories and the oscillatory behavior of solutions of complex differential equations. To apply the results to other areas of complex analysis.

    Growth of (alpha, beta, gamma)-order solutions to complex linear differential equations with analytic coefficients in the unit disc · 2026 · DOI

  • The study identifies a gap in the existing literature regarding the growth of solutions to complex linear differential equations with analytic coefficients in the unit disc. The paper aims to fill this gap by establishing new results and generalizing earlier works.

    Growth of (alpha, beta, gamma)-order solutions to complex linear differential equations with analytic coefficients in the unit disc · 2026 · DOI

  • Further study of the properties of the function φc(z, α) is suggested. Further study of the applications of the results to geometric functions is suggested. Further study of the extensions of the results to other classes of functions is suggested.

    Sharp Differential Subordinations for Cardioid Starlike Functions via Hypergeometric Techniques · 2026 · DOI

  • The paper identifies a gap in the study of differential subordinations for cardioid starlike functions. The paper identifies a need for the derivation of sufficient conditions for analytic functions to belong to a new Ma-Minda class of cardioid-starlike functions.

    Sharp Differential Subordinations for Cardioid Starlike Functions via Hypergeometric Techniques · 2026 · DOI

  • Future research can build on the results of the paper to establish new properties of conformal maps. Future research can explore the applications of the results in various contexts. Future research can investigate the relation between asymptotically symmetric embeddings and other concepts in complex analysis and geometry.

    On asymptotically symmetric embeddings and conformal maps · 2026 · DOI

  • The paper identifies a gap in the understanding of asymptotically symmetric embeddings and their relation to quasicircles. The paper identifies a need for a characterization of quasicircles in terms of asymptotically symmetric embeddings.

    On asymptotically symmetric embeddings and conformal maps · 2026 · DOI

Most-cited papers in Analytic and geometric function theory

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59 open questions have been extracted from the limitations and future-work passages of 466 Analytic and geometric function theory papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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